Properties

Label 7.6.a.b
Level 7
Weight 6
Character orbit 7.a
Self dual Yes
Analytic conductor 1.123
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 7 \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 7.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(1.12268673869\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)  \(=\)  \( q\) \( + ( 5 - \beta ) q^{2} \) \( + ( -6 + 6 \beta ) q^{3} \) \( + ( 7 - 9 \beta ) q^{4} \) \( + ( -4 - 10 \beta ) q^{5} \) \( + ( -114 + 30 \beta ) q^{6} \) \( + 49 q^{7} \) \( + ( 1 - 11 \beta ) q^{8} \) \( + ( 297 - 36 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 5 - \beta ) q^{2} \) \( + ( -6 + 6 \beta ) q^{3} \) \( + ( 7 - 9 \beta ) q^{4} \) \( + ( -4 - 10 \beta ) q^{5} \) \( + ( -114 + 30 \beta ) q^{6} \) \( + 49 q^{7} \) \( + ( 1 - 11 \beta ) q^{8} \) \( + ( 297 - 36 \beta ) q^{9} \) \( + ( 120 - 36 \beta ) q^{10} \) \( + ( 136 + 124 \beta ) q^{11} \) \( + ( -798 + 42 \beta ) q^{12} \) \( + ( -112 - 126 \beta ) q^{13} \) \( + ( 245 - 49 \beta ) q^{14} \) \( + ( -816 - 24 \beta ) q^{15} \) \( + ( -65 + 243 \beta ) q^{16} \) \( + ( 862 + 76 \beta ) q^{17} \) \( + ( 1989 - 441 \beta ) q^{18} \) \( + ( -1642 + 18 \beta ) q^{19} \) \( + ( 1232 + 56 \beta ) q^{20} \) \( + ( -294 + 294 \beta ) q^{21} \) \( + ( -1056 + 360 \beta ) q^{22} \) \( + ( 1328 - 568 \beta ) q^{23} \) \( + ( -930 + 6 \beta ) q^{24} \) \( + ( -1709 + 180 \beta ) q^{25} \) \( + ( 1204 - 392 \beta ) q^{26} \) \( + ( -3348 + 324 \beta ) q^{27} \) \( + ( 343 - 441 \beta ) q^{28} \) \( + ( 3474 - 252 \beta ) q^{29} \) \( + ( -3744 + 720 \beta ) q^{30} \) \( + ( 260 - 540 \beta ) q^{31} \) \( + ( -3759 + 1389 \beta ) q^{32} \) \( + ( 9600 + 816 \beta ) q^{33} \) \( + ( 3246 - 558 \beta ) q^{34} \) \( + ( -196 - 490 \beta ) q^{35} \) \( + ( 6615 - 2601 \beta ) q^{36} \) \( + ( 3386 - 540 \beta ) q^{37} \) \( + ( -8462 + 1714 \beta ) q^{38} \) \( + ( -9912 - 672 \beta ) q^{39} \) \( + ( 1536 + 144 \beta ) q^{40} \) \( + ( -3570 + 1092 \beta ) q^{41} \) \( + ( -5586 + 1470 \beta ) q^{42} \) \( + ( -3904 + 4788 \beta ) q^{43} \) \( + ( -14672 - 1472 \beta ) q^{44} \) \( + ( 3852 - 2466 \beta ) q^{45} \) \( + ( 14592 - 3600 \beta ) q^{46} \) \( + ( 7724 - 3748 \beta ) q^{47} \) \( + ( 20802 - 390 \beta ) q^{48} \) \( + 2401 q^{49} \) \( + ( -11065 + 2429 \beta ) q^{50} \) \( + ( 1212 + 5172 \beta ) q^{51} \) \( + ( 15092 + 1260 \beta ) q^{52} \) \( + ( 4630 + 208 \beta ) q^{53} \) \( + ( -21276 + 4644 \beta ) q^{54} \) \( + ( -17904 - 3096 \beta ) q^{55} \) \( + ( 49 - 539 \beta ) q^{56} \) \( + ( 11364 - 9852 \beta ) q^{57} \) \( + ( 20898 - 4482 \beta ) q^{58} \) \( + ( -22994 + 2050 \beta ) q^{59} \) \( + ( -2688 + 7392 \beta ) q^{60} \) \( + ( -34780 + 4806 \beta ) q^{61} \) \( + ( 8860 - 2420 \beta ) q^{62} \) \( + ( 14553 - 1764 \beta ) q^{63} \) \( + ( -36161 + 1539 \beta ) q^{64} \) \( + ( 18088 + 2884 \beta ) q^{65} \) \( + ( 36576 - 6336 \beta ) q^{66} \) \( + ( 11420 + 1944 \beta ) q^{67} \) \( + ( -3542 - 7910 \beta ) q^{68} \) \( + ( -55680 + 7968 \beta ) q^{69} \) \( + ( 5880 - 1764 \beta ) q^{70} \) \( + ( 46608 + 4200 \beta ) q^{71} \) \( + ( 5841 - 2907 \beta ) q^{72} \) \( + ( 6098 + 5256 \beta ) q^{73} \) \( + ( 24490 - 5546 \beta ) q^{74} \) \( + ( 25374 - 10254 \beta ) q^{75} \) \( + ( -13762 + 14742 \beta ) q^{76} \) \( + ( 6664 + 6076 \beta ) q^{77} \) \( + ( -40152 + 7224 \beta ) q^{78} \) \( + ( 33080 - 14904 \beta ) q^{79} \) \( + ( -33760 - 2752 \beta ) q^{80} \) \( + ( -24867 - 11340 \beta ) q^{81} \) \( + ( -33138 + 7938 \beta ) q^{82} \) \( + ( 66654 - 15750 \beta ) q^{83} \) \( + ( -39102 + 2058 \beta ) q^{84} \) \( + ( -14088 - 9684 \beta ) q^{85} \) \( + ( -86552 + 23056 \beta ) q^{86} \) \( + ( -42012 + 20844 \beta ) q^{87} \) \( + ( -18960 - 2736 \beta ) q^{88} \) \( + ( 31034 + 22208 \beta ) q^{89} \) \( + ( 53784 - 13716 \beta ) q^{90} \) \( + ( -5488 - 6174 \beta ) q^{91} \) \( + ( 80864 - 10816 \beta ) q^{92} \) \( + ( -46920 + 1560 \beta ) q^{93} \) \( + ( 91092 - 22716 \beta ) q^{94} \) \( + ( 4048 + 16168 \beta ) q^{95} \) \( + ( 139230 - 22554 \beta ) q^{96} \) \( + ( 14798 - 8820 \beta ) q^{97} \) \( + ( 12005 - 2401 \beta ) q^{98} \) \( + ( -22104 + 27468 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)  \(=\)  \(2q \) \(\mathstrut +\mathstrut 9q^{2} \) \(\mathstrut -\mathstrut 6q^{3} \) \(\mathstrut +\mathstrut 5q^{4} \) \(\mathstrut -\mathstrut 18q^{5} \) \(\mathstrut -\mathstrut 198q^{6} \) \(\mathstrut +\mathstrut 98q^{7} \) \(\mathstrut -\mathstrut 9q^{8} \) \(\mathstrut +\mathstrut 558q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 9q^{2} \) \(\mathstrut -\mathstrut 6q^{3} \) \(\mathstrut +\mathstrut 5q^{4} \) \(\mathstrut -\mathstrut 18q^{5} \) \(\mathstrut -\mathstrut 198q^{6} \) \(\mathstrut +\mathstrut 98q^{7} \) \(\mathstrut -\mathstrut 9q^{8} \) \(\mathstrut +\mathstrut 558q^{9} \) \(\mathstrut +\mathstrut 204q^{10} \) \(\mathstrut +\mathstrut 396q^{11} \) \(\mathstrut -\mathstrut 1554q^{12} \) \(\mathstrut -\mathstrut 350q^{13} \) \(\mathstrut +\mathstrut 441q^{14} \) \(\mathstrut -\mathstrut 1656q^{15} \) \(\mathstrut +\mathstrut 113q^{16} \) \(\mathstrut +\mathstrut 1800q^{17} \) \(\mathstrut +\mathstrut 3537q^{18} \) \(\mathstrut -\mathstrut 3266q^{19} \) \(\mathstrut +\mathstrut 2520q^{20} \) \(\mathstrut -\mathstrut 294q^{21} \) \(\mathstrut -\mathstrut 1752q^{22} \) \(\mathstrut +\mathstrut 2088q^{23} \) \(\mathstrut -\mathstrut 1854q^{24} \) \(\mathstrut -\mathstrut 3238q^{25} \) \(\mathstrut +\mathstrut 2016q^{26} \) \(\mathstrut -\mathstrut 6372q^{27} \) \(\mathstrut +\mathstrut 245q^{28} \) \(\mathstrut +\mathstrut 6696q^{29} \) \(\mathstrut -\mathstrut 6768q^{30} \) \(\mathstrut -\mathstrut 20q^{31} \) \(\mathstrut -\mathstrut 6129q^{32} \) \(\mathstrut +\mathstrut 20016q^{33} \) \(\mathstrut +\mathstrut 5934q^{34} \) \(\mathstrut -\mathstrut 882q^{35} \) \(\mathstrut +\mathstrut 10629q^{36} \) \(\mathstrut +\mathstrut 6232q^{37} \) \(\mathstrut -\mathstrut 15210q^{38} \) \(\mathstrut -\mathstrut 20496q^{39} \) \(\mathstrut +\mathstrut 3216q^{40} \) \(\mathstrut -\mathstrut 6048q^{41} \) \(\mathstrut -\mathstrut 9702q^{42} \) \(\mathstrut -\mathstrut 3020q^{43} \) \(\mathstrut -\mathstrut 30816q^{44} \) \(\mathstrut +\mathstrut 5238q^{45} \) \(\mathstrut +\mathstrut 25584q^{46} \) \(\mathstrut +\mathstrut 11700q^{47} \) \(\mathstrut +\mathstrut 41214q^{48} \) \(\mathstrut +\mathstrut 4802q^{49} \) \(\mathstrut -\mathstrut 19701q^{50} \) \(\mathstrut +\mathstrut 7596q^{51} \) \(\mathstrut +\mathstrut 31444q^{52} \) \(\mathstrut +\mathstrut 9468q^{53} \) \(\mathstrut -\mathstrut 37908q^{54} \) \(\mathstrut -\mathstrut 38904q^{55} \) \(\mathstrut -\mathstrut 441q^{56} \) \(\mathstrut +\mathstrut 12876q^{57} \) \(\mathstrut +\mathstrut 37314q^{58} \) \(\mathstrut -\mathstrut 43938q^{59} \) \(\mathstrut +\mathstrut 2016q^{60} \) \(\mathstrut -\mathstrut 64754q^{61} \) \(\mathstrut +\mathstrut 15300q^{62} \) \(\mathstrut +\mathstrut 27342q^{63} \) \(\mathstrut -\mathstrut 70783q^{64} \) \(\mathstrut +\mathstrut 39060q^{65} \) \(\mathstrut +\mathstrut 66816q^{66} \) \(\mathstrut +\mathstrut 24784q^{67} \) \(\mathstrut -\mathstrut 14994q^{68} \) \(\mathstrut -\mathstrut 103392q^{69} \) \(\mathstrut +\mathstrut 9996q^{70} \) \(\mathstrut +\mathstrut 97416q^{71} \) \(\mathstrut +\mathstrut 8775q^{72} \) \(\mathstrut +\mathstrut 17452q^{73} \) \(\mathstrut +\mathstrut 43434q^{74} \) \(\mathstrut +\mathstrut 40494q^{75} \) \(\mathstrut -\mathstrut 12782q^{76} \) \(\mathstrut +\mathstrut 19404q^{77} \) \(\mathstrut -\mathstrut 73080q^{78} \) \(\mathstrut +\mathstrut 51256q^{79} \) \(\mathstrut -\mathstrut 70272q^{80} \) \(\mathstrut -\mathstrut 61074q^{81} \) \(\mathstrut -\mathstrut 58338q^{82} \) \(\mathstrut +\mathstrut 117558q^{83} \) \(\mathstrut -\mathstrut 76146q^{84} \) \(\mathstrut -\mathstrut 37860q^{85} \) \(\mathstrut -\mathstrut 150048q^{86} \) \(\mathstrut -\mathstrut 63180q^{87} \) \(\mathstrut -\mathstrut 40656q^{88} \) \(\mathstrut +\mathstrut 84276q^{89} \) \(\mathstrut +\mathstrut 93852q^{90} \) \(\mathstrut -\mathstrut 17150q^{91} \) \(\mathstrut +\mathstrut 150912q^{92} \) \(\mathstrut -\mathstrut 92280q^{93} \) \(\mathstrut +\mathstrut 159468q^{94} \) \(\mathstrut +\mathstrut 24264q^{95} \) \(\mathstrut +\mathstrut 255906q^{96} \) \(\mathstrut +\mathstrut 20776q^{97} \) \(\mathstrut +\mathstrut 21609q^{98} \) \(\mathstrut -\mathstrut 16740q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
4.27492
−3.27492
0.725083 19.6495 −31.4743 −46.7492 14.2475 49.0000 −46.0241 143.103 −33.8970
1.2 8.27492 −25.6495 36.4743 28.7492 −212.248 49.0000 37.0241 414.897 237.897
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut -\mathstrut 9 T_{2} \) \(\mathstrut +\mathstrut 6 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(7))\).